%   ORIGINAL: 'h4/thm/list/LRC_MEM_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !y x ls e R. 'h4/const/list/LRC' R ls x y /\ 'h4/const/bool/IN' e ('h4/const/list/LIST_TO_SET' ls) ==> (?z t. R e z /\ 'h4/const/list/LRC' R t z y)
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !y x ls e R. 'h4/const/list/LRC' R ls x y /\ 'h4/const/bool/IN' e ('h4/const/list/LIST_TO_SET' ls) ==> (?z t. happ (happ R e) z /\ 'h4/const/list/LRC' R t z y)
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f7890,V_3f7886]: ![F, G]: (![X]: s(V_3f7886,happ(s(fun(V_3f7890,V_3f7886),F),s(V_3f7890,X))) = s(V_3f7886,happ(s(fun(V_3f7890,V_3f7886),G),s(V_3f7890,X))) => s(fun(V_3f7890,V_3f7886),F) = s(fun(V_3f7890,V_3f7886),G))).
fof('h4/thm/list/LRC_MEM_', conjecture, ![A]: ![Y, X, Ls, E, R]: ((p(s(bool,'h4/const/list/LRC'(s(fun(A,fun(A,bool)),R),s('h4/type/list/list'(A),Ls),s(A,X),s(A,Y)))) & p(s(bool,'h4/const/bool/IN'(s(A,E),s(fun(A,bool),'h4/const/list/LIST_TO_SET'(s('h4/type/list/list'(A),Ls))))))) => ?[Z, T]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,E))),s(A,Z)))) & p(s(bool,'h4/const/list/LRC'(s(fun(A,fun(A,bool)),R),s('h4/type/list/list'(A),T),s(A,Z),s(A,Y))))))).
